Move armature and object
I believe what you are looking to do is duplicate your armature, invert it (to create the "other" hand) and have it available with the same bone structure. If so, select the parent bone in object mode:



Press Shift D to duplicate it. I pressed Shift D + x to move the second hand along the X axis only:



You now have the second hand, although although the 'thumb' may be pointing outward. If your armature was like mine, you now have two armatures: Armature and Armature.001; all of the bone names will be unchanged.



Now, still in Object mode, scale the armature by pressing S (to scale), X to scale only along the X axis (in this instance, that is what is needed for my armature), and then -1. This will invert the armature along the X axis:

 

Now you have the same two hands, but with the thumbs both facing inward.  Scaling with a negative 1 scaling factor is a good way to create inverse versions of meshes and armatures.

Hope this is what you needed.